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Properties

Label 4-392e2-1.1-c0e2-0-0
Degree $4$
Conductor $153664$
Sign $1$
Analytic cond. $0.0382724$
Root an. cond. $0.442304$
Motivic weight $0$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s + 3·4-s − 4·8-s + 5·16-s + 2·25-s − 6·32-s − 4·50-s + 7·64-s − 4·67-s − 81-s + 6·100-s + 4·107-s − 2·121-s + 127-s − 8·128-s + 131-s + 8·134-s + 137-s + 139-s + 149-s + 151-s + 157-s + 2·162-s + 163-s + 167-s + 2·169-s + 173-s + ⋯
L(s)  = 1  − 2·2-s + 3·4-s − 4·8-s + 5·16-s + 2·25-s − 6·32-s − 4·50-s + 7·64-s − 4·67-s − 81-s + 6·100-s + 4·107-s − 2·121-s + 127-s − 8·128-s + 131-s + 8·134-s + 137-s + 139-s + 149-s + 151-s + 157-s + 2·162-s + 163-s + 167-s + 2·169-s + 173-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 153664 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 153664 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(153664\)    =    \(2^{6} \cdot 7^{4}\)
Sign: $1$
Analytic conductor: \(0.0382724\)
Root analytic conductor: \(0.442304\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 153664,\ (\ :0, 0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2995760116\)
\(L(\frac12)\) \(\approx\) \(0.2995760116\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$\Gal(F_p)$$F_p(T)$
bad2$C_1$ \( ( 1 + T )^{2} \)
7 \( 1 \)
good3$C_2^2$ \( 1 + T^{4} \)
5$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
11$C_2$ \( ( 1 + T^{2} )^{2} \)
13$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
17$C_2^2$ \( 1 + T^{4} \)
19$C_2^2$ \( 1 + T^{4} \)
23$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
29$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
31$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
37$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
41$C_2^2$ \( 1 + T^{4} \)
43$C_2$ \( ( 1 + T^{2} )^{2} \)
47$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
53$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
59$C_2^2$ \( 1 + T^{4} \)
61$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
67$C_1$ \( ( 1 + T )^{4} \)
71$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
73$C_2^2$ \( 1 + T^{4} \)
79$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
83$C_2^2$ \( 1 + T^{4} \)
89$C_2^2$ \( 1 + T^{4} \)
97$C_2^2$ \( 1 + T^{4} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.60850772798744020230300107710, −11.15767815449137423742214366434, −10.49256219553940935271304948290, −10.47532312828152730033100519341, −10.00930037858288919563303014591, −9.319089283845472391981756214243, −8.881218437890591878014609033370, −8.836505842614331460046430730780, −8.123806088468366138004013113671, −7.69499955236362975709200795848, −7.16453561690616187677470284519, −6.88355989854771910773220585407, −6.21258376867662354644552691581, −5.86342121267483289858218510497, −5.10615939355887923329439717519, −4.32698307489476940748726754712, −3.19680226230930867537077590072, −2.93752529136674474231738964203, −2.00986025874645470728683418307, −1.16546286981075352074177568114, 1.16546286981075352074177568114, 2.00986025874645470728683418307, 2.93752529136674474231738964203, 3.19680226230930867537077590072, 4.32698307489476940748726754712, 5.10615939355887923329439717519, 5.86342121267483289858218510497, 6.21258376867662354644552691581, 6.88355989854771910773220585407, 7.16453561690616187677470284519, 7.69499955236362975709200795848, 8.123806088468366138004013113671, 8.836505842614331460046430730780, 8.881218437890591878014609033370, 9.319089283845472391981756214243, 10.00930037858288919563303014591, 10.47532312828152730033100519341, 10.49256219553940935271304948290, 11.15767815449137423742214366434, 11.60850772798744020230300107710

Graph of the $Z$-function along the critical line